Anytime validity is free: inducing sequential tests

Abstract

Anytime valid sequential tests permit us to stop testing based on the current data, without invalidating the inference. Given a maximum number of observations N, one may believe this must come at the cost of power when compared to a conventional test that waits until all N observations have arrived. Our first contribution is to show that this is false: for any valid test based on N observations, we show how to construct an anytime valid sequential test that matches it after N observations. Our second contribution is that we may use the outcome of a [0,1]-valued test as a conditional significance level in subsequent testing, leading to an overall procedure that is valid at the original significance level. This shows that anytime validity and optional continuation are readily available in traditional testing, without requiring explicit use of e-values. We illustrate this by deriving the anytime valid sequentialized z-test and t-test, which at time N coincide with the traditional z-test and t-test. Finally, we characterize the Sequential Probability Ratio Test by invariance under test induction, and show that it is induced by the Neyman–Pearson test for a tiny level and huge N under an i.i.d. assumption.